This application claims the benefit of Korean Patent Application No. 2002-5649, filed Jan. 31, 2002, in the Korean Industrial Property Office, the disclosure of which is incorporated herein by reference.
1. Field of the Invention
The present invention relates to a method of compensating for image quality of a printing machine, and more particularly, to a method of compensating for image quality to provide a high quality image by effectively controlling a toner reproduction curve (TRC) to cope with environmental changes.
2. Description of the Related Art
An electrophotographic process used in a printing machine generally includes an initial step of charging a surface of a photoconductor. The charged surface of the photoconductor is exposed to light to form a latent electrostatic image in a specific image area. A develop unit controls a developing solution to adhere to the latent electrostatic image to develop the latent electrostatic image. The developed image is transferred to paper. The transferred image is fixed on the paper by a fixing roller.
In the step of charging the surface of the photoconductor, the photoconductor is charged with a uniform (constant) charge voltage so as to improve the quality of a printed image. Thus, the charge voltage charged on the photoconductor needs to be controlled to be uniform (constant). If the charge voltage is low, pollutants may occur in a non-image area. If the charge voltage is high, developed mass of the developing solution is changed. If the charge voltage is excessive, the photoconductor can become permanently damaged.
The charge voltage of the photoconductor is strengthened (modified) to a predetermined voltage, so-called xe2x80x9cexposure voltage,xe2x80x9d to form the latent electrostatic image in the exposure step. The developer unit has a developer bias so that a development voltage of the developing solution is higher than the exposure voltage of a portion on which the latent electrostatic image is formed, and lower than a non-image voltage of another portion of a photosensitive belt on which the latent electrostatic image is not formed. Due to this, a development step of adsorbing the developing solution to the latent electrostatic image may further be performed.
The developed mass of the developing solution absorbed in the latent electrostatic image is affected due to the exposure voltage and a development voltage as well as the charge voltage as described above.
A deviation (difference) between the development voltage and the exposure voltage becomes too big when the exposure voltage is low even though the uniform development voltage is applied. As a result, the adsorption of the developing solution increases. In contrast, the deviation between the development voltage and the exposure voltage is small when the exposure voltage is high even though the uniform development voltage is applied, thereby decreasing the adsorption of the developing solution. As a result, the developed image fades.
According to the above-described principle, when the photoconductor, which is charged with a predetermined charge voltage and a predetermined exposure voltage, is overcharged with the development voltage, a big deviation between the development voltage and the exposure voltage causes the developing solution to be excessively adsorbed on (attached to) the surface of the photoconductor. In contrast, the photoconductor is undercharged with the development voltage, a small deviation between the development voltage and the exposure voltage causes a relatively small amount of the adsorption of the developing solution on the photoconductor. As a result, the developed image fades.
Accordingly, efforts to develop an algorithm for properly controlling the charge voltage, the exposure voltage, and the development voltage have been made to compensate for the image quality. There was proposed a conventional method of compensating for the image quality by measuring the developed mass of an image on a photosensitive belt to control the printing machine since the above-mentioned three voltages affect the developed mass of the image.
FIG. 1 is a block diagram illustrating a conventional printing machine performing a method of controlling developed mass per unit area (DMA) to compensate for image errors so as to obtain a high quality image, which is disclosed in U.S. Pat. No. 5,749,021. The method suggests controlling the charge voltage, the exposure voltage, and the developer bias from internal process parameters, i.e., a discharge ratio, a cleaning voltage, and a development voltage.
The method of controlling the DMA improves the print quality by keeping the DMA under control in process control loops. Areas on which images are formed on a photoconductor are called xe2x80x9cimage areas,xe2x80x9d and test patches are generally prepared in a zone between the image areas of the photoconductor to be used for measuring the DMA. After the measured DMA is compared with a target value, errors are transmitted to a controller to control the internal process parameters so as to compensate for development errors. In other words, a grid voltage of a charger and an average beam power of an exposure system can be calculated from the internal process parameters to control subsystems of the printing machine.
Referring to FIG. 1, a level 1 controller 120 of a development system 140 transmits proper control signals Ug and Ul to an electrostatic charging and exposure system 122.
An electrostatic voltage sensor (ESV) measures a voltage of the electrostatic charging and exposure system 122 to obtain electrostatic and exposure voltage values Vh and Vl 124, respectively. The comparators 126a and 126b compare ESV sensor values Vh and Vl 124 with target values VhT and VlT 128 of the electrostatic and exposure voltage values Vh and Vl 124 to provide error signals Eh and El 129 to the level 1 controller 120. Gains of level 1 loops are obtained from the error signals Eh and El 129 to converge the voltage of the photoconductor to the target values VhT and VlT 128.
A level 2 controller 130 generates the target values VhT and VlT 128 to obtain the electrostatic and exposure voltage values Vh and Vl 124 through the level 1 controller 1 and the electrostatic charging and exposure system 122. Comparators 136a and 136b compare DMA sensor values Dl, Dm, and Dh 134, which are measured by a color toner density (CTD) sensor from the test patches prepared according to a toner area coverage, with target values DlT, DmT, and DhT 138, respectively, to provide error signals 139 to the level 2 controller 130. The level 2 controller 130 also generates a signal VTd to control a development system 132.
In other words, in the method of controlling the DMA, the DMA sensor values Dl, Dm, and Dh 134 measured by the CTD sensor are compared with the target values DlT, DmT, and DhT 138 to calculate deviations (differences) thereof. Thereafter, the calculated deviations are provided to the level 2 controller 130 to make the deviations linear with respect to the internal process parameters, i.e., the discharge ratio, the cleaning voltage, and the developer bias. Control parameters, i.e., the target values of the charging voltage, the exposure voltage, and the developer bias, are extracted from the linear discharge ratio, the cleaning voltage, and the developer bias to control the level 1 controller 120, the electrostatic charging and exposure system 122, and the development system 132. This control process will now be described with reference to FIG. 2.
FIG. 2 is a flowchart explaining the method of controlling the DMA in the printing machine of FIG. 1. Referring to FIG. 2, in the method of controlling the DMA, the DMA value is measured in step 101. Next, the measured DMA value is compared with the target DMA value to calculate a deviation thereof in step 103. If the deviation is smaller than a tolerance, a printing job is performed. If the deviation is greater than the tolerance, a control parameter displacement mass xcex94U is calculated by equation 1 in step 107. A new control parameter Unew is set by equation 2 to control the DMA in step 109.
xcex94U=xe2x88x92Gxc2x7xcex94Dxe2x80x83xe2x80x83(1)
UNEW=UOLD+xcex94Uxe2x80x83xe2x80x83(2)
The method of controlling the DMA as shown in FIG. 2 has a poor development control problem in the printing machine. The reason is that the DMA value measured by the CTD sensor contains noise components as well as the DMA value in the developer system. These noise components occur due to pollutants disposed on an organic photoconductive cell (OPC) or an intermediate transfer belt (ITB), a non-linearity of development characteristics, and other external disturbances.
When the control parameter displacement mass xcex94U of the control parameters is calculated to control the DMA, the deviation xcex94D between the target DMA value and the measured DMA value is multiplied by a gain matrix G to calculate the control parameter displacement mass xcex94U of the control parameters. Here, as seen in equation 3, the noise components are also multiplied to affect the control parameter displacement mass xcex94U of the control parameters.
xcex94U=Gxc2x7(xcex94D+n)xe2x80x83xe2x80x83(3)
Due to this multiplied noise components, if errors occur in the control parameter displacement mass xcex94U of the control parameter and if noise is great, the DMA cannot be controlled. If serious, the calculated control parameters become out of operative areas so as not to properly compensate for the image quality.
To solve the above and other problems, it is an object of the present invention to provide a method of compensating for image quality to obtain a high quality image by controlling a uniform toner reproduction curve (TRC) so as to exclude noise components that may be contained in measured DMA value.
Additional objects and advantageous of the invention will be set forth in part in the description which follows and, in part, will be obvious from the description, or may be learned by practice of the invention.
Accordingly, to achieve the above and other objects, a method of compensating for image quality of an printing machine having a color toner density sensor, which is provided over a photosensitive belt, includes receiving light reflected from test patches each having a different toner area coverage and converting the received light to an electrical signal to control a developer bias VB and a grid voltage VG. The method includes: (a) comparing a toner reproduction curve (TRC) "PHgr"(k) value measured by the color toner density sensor with a target TRC value "PHgr"R to obtain a deviation xcex94"PHgr"; (b) calculating a variation xcex94VB of the developer bias VB from a Jacobian matrix (JB) of a measured developer bias VBO to calculate a developer bias control parameter VBN and determining a measured grid voltage VGO as a grid voltage control parameter VGN if the deviation (xcex94"PHgr") is greater than tolerance xcex94"PHgr"T; (c) obtaining a backplating vector VBP from the grid voltage control parameter VGN and the developer bias control parameter VBN; and (d) comparing the backplating vector VBP with a critical value VT to set the grid voltage control parameter VGN and the developer bias control parameter VBN as new control parameters VGN and VBN to control a TRC "PHgr", the developer bias VB, and the grid voltage VG.
In operation (a), the deviation xcex94"PHgr" satisfies equation 4.
xcex94"PHgr"=|"PHgr"Oxe2x88x92"PHgr"T|xe2x80x83xe2x80x83(4)       where    ⁢          xe2x80x83        ⁢          Φ      o        =                    [                                                            Φ                OL                                                                                        Φ                OM                                                                                        Φ                OH                                                    ]            ⁢      and      ⁢              xe2x80x83            ⁢              Φ        T              =                  [                                                            Φ                TL                                                                                        Φ                TM                                                                                        Φ                TH                                                    ]            ⁢              xe2x80x83            .      
Operation (b) includes: (b-1) calculating the variation xcex94VB of the developer bias VB, which satisfies equation 5, from the Jacobian matrix JB of the measured developer bias VB;
xcex94VB=xe2x88x92GBxc2x7xcex94"PHgr"xe2x80x83xe2x80x83(5)       where    ⁢          xe2x80x83        ⁢          G      B        =      
    ⁢                                          (                                          J                B                T                            ·                              J                B                                      )                                -            1                          ·                  J          B          T                    ⁢              xe2x80x83            ⁢      and      ⁢              xe2x80x83            ⁢      where      ⁢              xe2x80x83            ⁢              J        B              =          "AutoLeftMatch"                        [                                                                      J                  BL                                                                                                      J                  BM                                                                                                      J                  BH                                                              ]                =                              [                                                                                                      2                      ⁢                                              A                        L                        xe2x80x2                                            ⁢                                              V                        B                                                              +                                                                  F                        L                        xe2x80x2                                            ⁢                                              V                        G                                                              +                                          C                      L                      xe2x80x2                                                                                                                                                              2                      ⁢                                              A                        M                        xe2x80x2                                            ⁢                                              V                        B                                                              +                                                                  F                        M                        xe2x80x2                                            ⁢                                              V                        G                                                              +                                          C                      M                      xe2x80x2                                                                                                                                                              2                      ⁢                                              A                        H                        xe2x80x2                                            ⁢                                              V                        B                                                              +                                                                  F                        H                        xe2x80x2                                            ⁢                                              V                        G                                                              +                                          C                      H                      xe2x80x2                                                                                            ]                    =                      [                                                                                                      ∂                                              Φ                        OL                                                                                    ∂                                              V                        B                                                                                                                                                                                    ∂                                              Φ                        OM                                                                                    ∂                                              V                        B                                                                                                                                                                                    ∂                                              Φ                        OH                                                                                    ∂                                              V                        B                                                                                                                  ]                              
(b-2) setting a new developer bias control parameter VBN, which satisfies equation 6, from the variation xcex94VB of the developer bias VB; and
VBN=VB+xcex94VBxe2x80x83xe2x80x83(6)
(b-3) determining the measured grid voltage VG as a new grid voltage VGN.
In operation (c), the backplating vector VBP, which satisfies equation 7, is calculated from the new grid voltage control parameter VGN and the new developer bias control parameter VBN.
VBP=VGNxe2x88x92VBNxe2x80x83xe2x80x83(7)
Operation (d) includes: (d-1) determining the new grid voltage control parameter VGN and the new developer bias control parameter VBN in operation (c) as control parameters if the backplating vector VBP is greater than the critical value VT; (d-2) calculating the developer bias control parameter VBN and the grid voltage control parameter VGN, which satisfy equation 8, from the Jacobian matrix JB of the measured developer bias VBO, Jacobina matrix JG of the measured grid voltage VGO, and a TRC control parameter C if the backplating vector VBP is smaller than the critical value VT;
xe2x80x83VBN=VB+xcex94VB
VGN=VG+xcex94VGxe2x80x83xe2x80x83(8)
where xcex94VB=xe2x88x92GBxc2x7xcex94"PHgr"xc2x7(I+C) and xcex94VG=xe2x88x92GGxc2x7xcex94"PHgr"xc2x7C, where GG=(JGTxc2x7JG)xe2x88x921xc2x7JGT, and       where    ⁢          xe2x80x83        ⁢          J      G        =            [                                                  J              GL                                                                          J              GM                                                                          J              GH                                          ]        =                  [                                                                              2                  ⁢                                      B                    L                    xe2x80x2                                    ⁢                                      V                    G                                                  +                                                      E                    L                    xe2x80x2                                    ⁢                                      V                    V                                                  +                                  D                  L                  xe2x80x2                                                                                                                          2                  ⁢                                      B                    M                    xe2x80x2                                    ⁢                                      V                    G                                                  +                                                      E                    M                    xe2x80x2                                    ⁢                                      V                    B                                                  +                                  D                  M                  xe2x80x2                                                                                                                          2                  ⁢                                      B                    H                    xe2x80x2                                    ⁢                                      V                    G                                                  +                                                      E                    H                    xe2x80x2                                    ⁢                                      V                    B                                                  +                                  D                  H                  xe2x80x2                                                                    ]            =              [                                                                              ∂                                      Φ                    OL                                                                    ∂                                      V                    G                                                                                                                                            ∂                                      Φ                    OM                                                                    ∂                                      V                    G                                                                                                                                            ∂                                      Φ                    OH                                                                    ∂                                      V                    G                                                                                      ]            
(d-3) increasing the TRC control parameter C by an increment a so as to satisfy equation 9 if the backplating vector VBP is smaller than the critical value VT;
C=C+xcex1xe2x80x83xe2x80x83(9)
(d-4) repeating operation (d-3) until the backplating vector VBP becomes greater than the critical value VT; and (d-5) determining the new grid voltage control parameter VGN and the new developer bias control parameter VBN as the new control parameters VGN and VBN when the backplating vector VBP is greater than the critical value VT.
To achieve the above and other objects, a method of compensating for the image quality of an printing machine having a color toner density sensor, which is provided over a photosensitive belt, includes receiving light reflected from test patches with different toner area coverages and converting the received light to an electrical signal to control a develop bias VB and a grid voltage VG. The method includes: (a) comparing a toner reproduction curve "PHgr"(k) measured by the color toner density sensor with a reference TRC "PHgr"R to obtain a deviation xcex94"PHgr"; (b) calculating a variation xcex94VB of a developer bias VB(k) from a Jacobian matrix JB of a measured developer bias VB(k) to calculate a new developer bias control parameter VB(k+1) and determining a measured grid voltage VG(k) as a new grid voltage control parameter VG(k+1) if the deviation xcex94"PHgr" is not less than a tolerant deviation xcex94"PHgr"T; and (c) obtaining a backplating vector VBP from a difference between the grid voltage control parameter VG(k+1) and the developer bias control parameter VB(k+1).
The method further includes (d) initializing a control parameter a, comparing the backplating vector VBP with a minimum critical value VTmin, performing operation (e) if the backplating vector VBP is smaller than the minimum critical value VTmin, and performing operation (g) if the backplating vector VBP is greater than the minimum critical value VTmin; (e) increasing the control parameter xe2x80x9caxe2x80x9d by an increment xe2x80x9cxcex1xe2x80x9d, setting the developer bias and grid voltage control parameters VG(k+1) and VB(k+1) based on an amount of the deviation xcex94"PHgr", obtaining the backplating vector VBP from the difference between the grid voltage control parameter VG(k+1) and the developer bias control parameter VB(k+1), and performing operation (f); (f) repeating operations (e) if the backplating vector VBP is smaller than the minimum critical value VTmin, and repeating operations (a) through (e) if the backplating vector VBP is greater than the minimum critical value VTmin; (g) increasing the control parameter xe2x80x9caxe2x80x9d by the increment xe2x80x9cxcex1xe2x80x9d, setting control parameters VG(k+1) and VB(k+1) based on an amount of the deviation xcex94"PHgr", obtaining the backplating vector VBP from the difference between the control parameters VG(k+1) and VB(k+1), and performing operation (h); (h) repeating operation (g) if the backplating vector VBP is greater than a maximum critical value VTmax, and repeating operations (a) through (g) if the backplating vector VBP is smaller than a maximum critical value VTmax.
In operation (a), the deviation xcex94"PHgr" satisfies equation 10.
xcex94"PHgr"="PHgr"(k)xe2x88x92"PHgr"Rxe2x80x83xe2x80x83(10)       where    ⁢          xe2x80x83        ⁢          Φ      ⁡              (        k        )              =                    [                                                                              Φ                  ⁡                                      (                    k                    )                                                  L                                                                                                          Φ                  ⁡                                      (                    k                    )                                                  M                                                                                                          Φ                  ⁡                                      (                    k                    )                                                  H                                                    ]            ⁢      and      ⁢              xe2x80x83            ⁢              Φ        R              =                  [                                                            Φ                RL                                                                                        Φ                RM                                                                                        Φ                RH                                                    ]            ⁢              xe2x80x83            .      
Operation (b) includes: (b-1) calculating the variation xcex94VB of the developer bias VB(k), which satisfies equation 11, from the Jacobian matrix JB of the measured developer bias VB(k);
xcex94VB=xe2x88x92GBxc2x7xcex94"PHgr"xe2x80x83xe2x80x83(11)                     Where        ⁢                  xe2x80x83                ⁢                  G          B                    =                                                  (                                                J                  B                  T                                ·                                  J                  B                                            )                                      -              1                                ·                      J            B            T                          ⁢                  xe2x80x83                ⁢        where              ⁢          xe2x80x83                  J      B        =                  [                                                            J                BL                                                                                        J                BM                                                                                        J                BH                                                    ]            =                        [                                                                                          2                    ⁢                                          A                      L                                        ⁢                                          V                      B                                                        +                                                            E                      L                                        ⁢                                          V                      G                                                        +                                      C                    L                                                                                                                                            2                    ⁢                                          A                      M                                        ⁢                                          V                      B                                                        +                                                            E                      M                                        ⁢                                          V                      G                                                        +                                      C                    M                                                                                                                                            2                    ⁢                                          A                      H                                        ⁢                                          V                      B                                                        +                                                            E                      H                                        ⁢                                          V                      G                                                        +                                      C                    H                                                                                ]                =                  [                                                                                          ∂                                                                  Φ                        ⁡                                                  (                          k                          )                                                                    L                                                                            ∂                                          V                      B                                                                                                                                                                ∂                                                                  Φ                        ⁡                                                  (                          k                          )                                                                    M                                                                            ∂                                          V                      B                                                                                                                                                                ∂                                                                  Φ                        ⁡                                                  (                          k                          )                                                                    H                                                                            ∂                                          V                      B                                                                                                    ]                    
(b-2) setting a new developer bias control parameter VB(k+1), which satisfies equation 12, from the variation of the developer bias control input value xcex94VB of the developer bias VB; and
VB(k+1)=VB(k)+xcex94VBxe2x80x83xe2x80x83(12)
(b-3) determining the measured grid voltage VG(k) as a new grid voltage control parameter VGN(k+1).
In operation (c), the backplating vector VBP, which satisfies equation 13, is calculated from the grid voltage control parameter VG(k+1) and the developer bias control parameter VB(k+1).
VBP=VG(k+1)xe2x88x92VB(k+1)xe2x80x83xe2x80x83(13)
In operation (d), the control parameter xe2x80x9caxe2x80x9d is initialized as xe2x80x9c0xe2x80x9d.
Operation (e) includes: (e-1) incrementing the control parameter a by an increment xe2x80x9cxcex1xe2x80x9d according to equation 14;
a=a+xcex1xe2x80x83xe2x80x83(14)
(e-2) setting the control parameters VG(k+1) and VB(k+1) that satisfy equation 15 to obtain the backplating vector VBP from the difference between the grid voltage control parameter VG(k+1) and the developer bias control parameter VB(k+1) if the deviation xcex94"PHgr" is negative and then going to operation (t); and
VB(k+1)=VB(k)+xcex94VB
VG(k+1)=VG(k)+xcex94VGxe2x80x83xe2x80x83(15)
VBP=VG(k+1)xe2x88x92VB(k+1)
where xcex94VB=xe2x88x92GBxc2x7xcex94"PHgr"xc2x7(1+a) and xcex94VG=GGxc2x7xcex94"PHgr"xc2x7a, where GG=(JGTxc2x7JG)xe2x88x921xc2x7JGT, and where       J    G    =            [                                                  J              GL                                                                          J              GM                                                                          J              GH                                          ]        =                  [                                                                              2                  ⁢                                      B                    L                                    ⁢                                      V                    G                                                  +                                                      E                    L                                    ⁢                                      V                    B                                                  +                                  D                  L                                                                                                                          2                  ⁢                                      B                    M                                    ⁢                                      V                    G                                                  +                                                      E                    M                                    ⁢                                      V                    B                                                  +                                  D                  M                                                                                                                          2                  ⁢                                      B                    H                                    ⁢                                      V                    G                                                  +                                                      E                    H                                    ⁢                                      V                    G                                                  +                                  D                  H                                                                    ]            =                        [                                                                                          ∂                                                                  Φ                        ⁡                                                  (                          k                          )                                                                    L                                                                            ∂                                          V                      G                                                                                                                                                                ∂                                                                  Φ                        ⁡                                                  (                          k                          )                                                                    M                                                                            ∂                                          V                      G                                                                                                                                                                ∂                                                                  Φ                        ⁡                                                  (                          k                          )                                                                    H                                                                            ∂                                          V                      G                                                                                                    ]                ⁢                  xe2x80x83                ,            
(e-3) setting the control parameters VG(k+1) and VB(k+1) that satisfy equation 16 to obtain the backplating vector VBP from the difference between the grid voltage control parameter VG(k+1) and the developer bias control parameter VB(k+1) if the deviation xcex94"PHgr" is positive and then going to operation (f); and
VB(k+1)=VB(k)+xcex94VB
VG(k+1)=VG(k)+xcex94VGxe2x80x83xe2x80x83(16)
VBP=VG(k+1)xe2x88x92VB(k+1)
where xcex94VB=xe2x88x92GBxc2x7xcex94"PHgr"xc2x7(1xe2x88x92a) and xcex94VG=xe2x88x92GGxc2x7xcex94"PHgr"xc2x7a, where GG=(JGTxc2x7JG)xe2x88x921xc2x7JGT, and       where    ⁢          xe2x80x83        ⁢          J      G        =            [                                                  J              GL                                                                          J              GM                                                                          J              GH                                          ]        =                  [                                                                              2                  ⁢                                      B                    L                                    ⁢                                      V                    G                                                  +                                                      E                    L                                    ⁢                                      V                    B                                                  +                                  D                  L                                                                                                                          2                  ⁢                                      B                    M                                    ⁢                                      V                    G                                                  +                                                      E                    M                                    ⁢                                      V                    B                                                  +                                  D                  M                                                                                                                          2                  ⁢                                      B                    H                                    ⁢                                      V                    G                                                  +                                                      E                    H                                    ⁢                                      V                    G                                                  +                                  D                  H                                                                    ]            =                        [                                                                                          ∂                                                                  Φ                        ⁡                                                  (                          k                          )                                                                    L                                                                            ∂                                          V                      G                                                                                                                                                                ∂                                                                  Φ                        ⁡                                                  (                          k                          )                                                                    M                                                                            ∂                                          V                      G                                                                                                                                                                ∂                                                                  Φ                        ⁡                                                  (                          k                          )                                                                    H                                                                            ∂                                          V                      G                                                                                                    ]                ⁢                  xe2x80x83                .            
Operation (g) includes: (g-1) incrementing the control parameter xe2x80x9caxe2x80x9d by the increment xcex1; (g-2) setting the control parameters VG(k+1) and VB(k+1) that satisfy equation 17 to obtain the backplating vector VBP from a difference between the grid voltage control parameter VG(k+1) and the developer bias control parameter VB(k+1) if the deviation (xcex94"PHgr") is negative and then going to operation (h); and
xe2x80x83VB(k+1)=VB(k)+xcex94VB
VG(k+1)=VG(k)+xcex94VGxe2x80x83xe2x80x83(17)
VBP=VG(k+1)xe2x88x92VB(k+1)
where xcex94VB=xe2x88x92GBxc2x7xcex94"PHgr"xc2x7(1xe2x88x92a) and xcex94VG=GGxc2x7xcex94"PHgr"xc2x7a, where GG=(JGTxc2x7JG)xe2x88x921xc2x7JGTand where       J    G    =            [                                                  J              GL                                                                          J              GM                                                                          J              GH                                          ]        =                  [                                                                              2                  ⁢                                      B                    L                                    ⁢                                      V                    G                                                  +                                                      E                    L                                    ⁢                                      V                    B                                                  +                                  D                  L                                                                                                                          2                  ⁢                                      B                    M                                    ⁢                                      V                    G                                                  +                                                      E                    M                                    ⁢                                      V                    B                                                  +                                  D                  M                                                                                                                          2                  ⁢                                      B                    H                                    ⁢                                      V                    G                                                  +                                                      E                    H                                    ⁢                                      V                    G                                                  +                                  D                  H                                                                    ]            =              [                                                                              ∂                                                            Φ                      ⁡                                              (                        k                        )                                                              L                                                                    ∂                                      V                    G                                                                                                                                            ∂                                                            Φ                      ⁡                                              (                        k                        )                                                              M                                                                    ∂                                      V                    G                                                                                                                                            ∂                                                            Φ                      ⁡                                              (                        k                        )                                                              H                                                                    ∂                                      V                    G                                                                                      ]            
(g-3) setting the control parameters VG(k+1) and VB(k+1) that satisfy equation 18 obtain the backplating vector VBP from the difference between the grid voltage control parameter VG(k+1) and the developer bias control parameter VB(k+1) if the deviation xcex94"PHgr" is positive and then going to operation (f); and
VB(k+1)=VB(k)+xcex94VB
VG(k+1)=VG(k)+xcex94VGxe2x80x83xe2x80x83(18)
VBP=VG(k+1)xe2x88x92VB(k+1)
where xcex94VB=xe2x88x92GBxc2x7xcex94"PHgr"xc2x7(1xe2x88x92a) and xcex94VG=+GGxc2x7xcex94"PHgr"xc2x7a, where GG=(JGTxc2x7JG)xe2x88x921xc2x7JGT, and       where    ⁢          xe2x80x83        ⁢          J      G        =            [                                                  J              GL                                                                          J              GM                                                                          J              GH                                          ]        =                  [                                                                              2                  ⁢                                      B                    L                                    ⁢                                      V                    G                                                  +                                                      E                    L                                    ⁢                                      V                    B                                                  +                                  D                  L                                                                                                                          2                  ⁢                                      B                    M                                    ⁢                                      V                    G                                                  +                                                      E                    M                                    ⁢                                      V                    B                                                  +                                  D                  M                                                                                                                          2                  ⁢                                      B                    H                                    ⁢                                      V                    G                                                  +                                                      E                    H                                    ⁢                                      V                    G                                                  +                                  D                  H                                                                    ]            =                        [                                                                                          ∂                                                                  Φ                        ⁡                                                  (                          k                          )                                                                    L                                                                            ∂                                          V                      G                                                                                                                                                                ∂                                                                  Φ                        ⁡                                                  (                          k                          )                                                                    M                                                                            ∂                                          V                      G                                                                                                                                                                ∂                                                                  Φ                        ⁡                                                  (                          k                          )                                                                    H                                                                            ∂                                          V                      G                                                                                                    ]                ⁢                  xe2x80x83                .            
In the present invention, a high quality image can be provided by uniformly maintaining the toner reproduction curve in spite of external disturbances and changes of internal systems to uniformly control the developed mass per unit area regardless of noise components contained in the developed mass per unit area.